STOCK MARKET RETURN ANOMALIES

The literature on stock market anomalies is difficult to summarize briefly, because many contradictory results are reported and frequently the origin of differences in empirical results is not very clear. Note that there is a definite bias in the review presented below. I am interested in how the various results may still fit the general efficient market model, and where results show the limitations of some of the auxiliary (simplifying) assumptions of the textbook model.

The literature contains a sufficient number of suggestions and empirical results that point to possible solutions to the many puzzles. I find that there is little reason to declare a definite default on the fundamental economic hypotheses of rational behavior (on aggregate or market level) and efficient markets (with proper consideration of real world constraints). But you need to be willing to go beyond the simplistic, stylized textbook models and the simple assumptions maintained in much of the mainstream literature.

CROSS-SECTION OF STOCK RETURNS

THE SIZE EFFECT

The size effect refers to the negative relation between stock returns and the market value of equity of a firm. Banz (1981) was the first to document this phenomenon for U.S. stocks (see also Reinganum, 1981). Banz found that the coefficient on size has more explanatory power than the coefficient on the CAPM beta in describing the cross section of returns. The size effect has been reproduced for numerous sample periods and for most major securities markets around the world (Hawawini and Keim, 2000).

Booth and Keim (2000, Table 1 NYSE-AMEX-NASDAQ stocks) estimate the small firm (Decile10)-large firm (Decile1) effect to be statistically significant at 0.81% per month for all months 1966-1981. Horowitz, Loughran and Savin (2000, Table 1 NYSE-AMEX-NASDAQ stocks) estimate 1.09% per month for all months 1963-1981.

• The size effect is only a January effect; no size effect exists in the average returns of other months of the year (Hawawini and Keim, 2000, Figure 1 Table 5 NYSE-AMEX stocks 1962-1994; Booth and Keim, 2000 Table 1 and 2 NYSE-AMEX-NASDAQ 1926-1981, 1982-1995).
• The size effect is exaggerated due to delisting bias in the CRSP database. Correction for negative delisting returns eliminates the size effect in NASDAQ stocks (Shumway and Warther, 1999) although not in NYSE-AMEX stocks (Shumway, 1997).
• The size effect is exaggerated due to the bid-ask bounce and rebalancing effect in calculated short-run (daily, monthly) returns (Conrad and Kaul, 1993).
• Boynton and Oppenheimer (2006) calculate that more than 40% of the commonly reported size premium in monthly returns can be attributed to delisting and bid-ask spread bias: reducing it from 1.36% to 0.73% per month for the full-sample 1926-2002 period. Using geometric returns for long-run returns reduces the size premium to 0.33% (no longer significant).
• The small price effect is driven by very small price stocks (penny stocks) that sometimes show extremely large returns (for example, 500% to 2000%) (Horowitz, Loughran and Savin, 2000). Eliminating small price or small cap stocks (less than $5mln) eliminates the size effect (Horowitz, Loughran and Savin, 2000 Table 2). Some of these returns may be related to CRSP database errors, for example failures to adjust for (reverse) stock splits and stock conversions following bankruptcy reorganizations (Bhardwaj and Brooks, 1992 footnote 8).
• The size effect is unreliable. In the US the average size effect over all months has disappeared after 1981 (Booth and Keim, 2000 Table 2; Horowitz, Loughran and Savin, 2000 Table 1 and Figure A) and was never consistent to begin with (Booth and Keim, 2000 Table 1). The disappearance of the size effect is mirrored in the decline of the January effect in small stocks. The size effect is driven by brief periods of extraordinary high returns, such as the 1975-1983 period in the US (Siegel, 1998 Figure 6-1 1925-1997) or extraordinary low returns, such as the 1969-1973 period (Brown, Kleidon and Marsh, 1983). The average size effect also disappeared in the UK after 1989 and in fact turned into a negative effect (Dimson and Marsh, 2000 Table 2).
• The size effect is actually a small or low price effect; other indicators of size such as book value, sales, number of employees, etc. have no explanatory power (Berk 1995, 2000)
• A relationship between low price and higher returns is not in any way anomalous. Higher expected/required returns will result in a low price because there needs to be room future future price appreciation that creates the higher return (Berk 1995, 2000)
• Failure to find strong effects of the traditional CAPM beta may be due to errors-in-variables bias (Kim, 1995, 1997; see also Chan and Chen, 1988). The CAPM-beta effect also depends on the return measurement interval (Hand, Kothari and Wasley, 1993; Kothari, Shanken and Sloan, 1995)
• Size portfolios (as well as B/M portfolios) are highly correlated with the KMV distance-to-default risk measure (Vassalou/Xing, 2004)
• Liquidity risk proxied by the bid-ask spread variable is an additional risk-factor (Amihud and Mendelson, 1989)
• The size effect is irrelevant with respect to the efficient market hypothesis
• Bid-ask bounce effect: if on average prices in December reflect sell transactions at bid prices and prices in January reflect buy transactions at ask prices (see the January effect), the average high return over January simply reflects the high bid-ask spread for small or low price stocks. There is no trading opportunity in such observed data.
• Including appropriate transaction costs for the relevant stocks eliminates any profits from frequent trading strategies based on the usual daily or monthly return studies (for example Stoll and Whaley, 1983; Schultz, 1983; Bhardwaj and Brooks, 1992; Al-Rajoub and Hassan, 2004).
• Long horizon buy-hold returns suffer less from trading costs (Stoll and Whaley, 1983; Schultz, 1983), but one needs to be very careful in calculating the appropriate holding period returns (avoiding the portfolio rebalancing problem and other problems of aggregating monthly returns (for example, Conrad and Kaul, 1993; Roll, 1983; Blume and Stambaugh, 1983). Statistical inference is difficult due to overlapping observations and/or small sample problems.
• Investment funds such as the DFA9-10 Small Company Portfolio find it impossible to capture the size effect, as a result of practical constraints on investing in the smallest capitalization (less than $10mln) and smallest price stocks (less than $2) (Booth and Keim, 2000 Table 2).

CASE CLOSED?

THE VALUE vs. GROWTH EFFECT

Basu (1977, 1983) found that firms with high E/P ratios earned positive abnormal returns relative to the CAPM model. Subsequent research established similar effects of other accounting based ratios such as cash flow (CF/P) and book value of equity (BM). Current consensus seems to be that earnings and cash flow effects are subsumed by the book-to-market ratio and firm size (Fama and French, 1993).

• The value effect based on CF/P, E/P, BM is only a January effect. No value effect exists in the average returns of the other months of the year (Hawawini and Keim, 2000, Figure 1 Table 5 NYSE-AMEX stocks 1962-1994)
• The BM effect is exaggerated due to delisting bias and rebalancing effects.
• Boynton and Oppenheimer (2006) calculate that 25% of the commonly reported BM premium (BM10-BM1) in monthly returns can be attributed to delisting and rebalancing bias: reducing it from 1.32% to 1.01% (still significant) per month for the full-sample 1966-2002 period.
• The value effect is irrelevant with respect to the efficient market hypothesis.
• Including appropriate transaction costs for the relevant stocks eliminates any profits from value trading strategies based on an annual buy-hold return study (for example Agarwal and Wang, 2007).
• Returns of the DFA US 6-10 value portfolio, that focuses on small firms with high B/M ratios show that the value effect cannot be captured in real time. The abnormal return for 1994-2002 is -0.2% (not significant) (Schwert, 2003 Table1).

THE MOMENTUM EFFECT

Jegadeesh and Titman (1993) found that recent past winners (stocks with high returns in the past 12 months) outperform recent past losers. The momentum effect for the winner-loser portfolio is estimated to be around 1% per month, after correction for CAMP-beta or Fama-French 3-factor model (Schwert, 2003 Table 4). Whereas Schwert (2003) finds the momentum effect relatively robust across subsamples in 1926-2001, Kothari, Shanken and Sloan (1995) find the momentum effect unreliable with in the post-1962 period winners earning negative abnormal returns and losers earning positive abnormal returns.

• The momentum effect in itself is not an anomaly. In a cross-section of stocks where average, expected returns are dispersed - presumably due to differences in risk premiums - high returns and low returns tend to persist. Momentum is found to be strong and highly significant, but this is fundamental (Conrad and Kaul, 1998; Lo and MacKinlay, 1990). One needs to separate any anomalous momentum effect (unwarranted time series predictability) from the fundamental momentum effect (cross-section dispersion effect and time-varying risk premiums), which may be very difficult.
• Bulkley and Nawosah (2008) report that simply correcting for individual stock's sample average returns eliminates the momentum effect. This in contrast to Jegadeesh and Titman (2002) who find that demeaning returns in their sample does not eliminate momentum. Their residual momentum effect appears to be related to some special feature of NASDAQ stocks.
• Approx. 40% of the momentum effect in NYSE-AMEX-Nasdaq stocks reflects the delisting returns of 'bankruptcy' firms (Eisdorfer, 2008 Table 3). In order to capture this return, momentum investors must keep a short position in these stocks after the delisting day, which is not likely. (Note: Shumway 1997 and Shumway and Warther 1999 show that delisting returns reported in the CRSP database need to be evaluated carefully.)
• Momentum profitability is large and significant among firms with low credit ratings, but nonexistent among high-grade firms. Momentum is attributable to continuation among low-grade loser firms that experience negative returns following rating downgrades. Loser stocks exhibit lower analyst following, negative analyst forecast revisions and negative earnings surprises following the portfolio formation date (Avramov, Chordia, Jostova and Philipov, 2006).
• The cost of frequent trading in the momentum portfolio, together with the disproportionately high costs of precisely the stocks on which the anomaly relies, eliminates any perceived arbitrage profits (Lesmond, Schill and Zhou, 2004).

THE LONG-HORIZON WINNER-LOSER EFFECT (a.k.a. return reversal, overreaction, contrarian strategy)

• The Winner-Loser contrarian effect is exaggerated due to delisting bias and rebalancing effects.
• Boynton and Oppenheimer (2006) calculate that 25% of the reported winner-loser premium (C1-C20) in monthly returns on a 36 month horizon can be attributed to delisting and rebalancing bias: reducing it from 0.78% to 0.60% (still significant) per month (NYSE, 1930-2001). Shumway (1997) shows that the delisting effect is small, but using buy-hold returns instead of eliminates the winner-loser effect in a replication of the De Bondt and Thaler results (NYSE, 1926-1989).
• Ball, Kothari and Shanken (1995) show a 50% decline in the winner-loser premium after adjusting for small price stock effects (increasing prices by one tick $1/8) and changing the portfolio formation from December to June.
• Including data from the U.S. Great Depression also tends to bias the mean reversion effect (Kim/Nelson/Startz, 1991).
• Although still open to debate, a number of studies have found that the winner-loser contrarian effect disappears with a correction for normal returns based on the Fama-French 3-factor model (for example, Fama and French, 1995, 1996; Clements, Drew, Reedman and Veeraraghavan, 2007). The FF 3-factor model has been criticized for its lack of theoretical foundation in risk models. However, research has shown a strong link between measures of default risk and small company size and company B/M-ratios (Vassalou/Xing, 2004). Others have also shown a strong relationship between the big-small and high-low factor portfolios and the business cycle, but so far fail to provide strong arguments for the type of risk that this would represent.

SEASONAL PATTERNS IN STOCK RETURNS

THE JANUARY EFFECT

The January effect refers to the tendency for stock market returns to be higher in January than in any of the other months (Rozeff and Kinney, 1976). The January effect is particularly strong in small size stocks (Keim, 1983), but is also present in large stocks, and also present in value stocks based on CF/P, E/P, P/B (Hawawini and Keim, 2000 Figure 1). The January effect is mainly located in the first 2 weeks of January.

Booth and Keim (2000, Table 1 NYSE-AMEX-NASDAQ stocks) estimate the small firm (Decile10)-large firm (Decile1) January effect to be statistically significant at 9.72% during 1966-1981 and 4.48% for 1982-1995. Note that the estimates of the small-firm January effect are variable over time, with high returns for 1926-1945, lower returns for 1946-1965, higher returns for 1966-1981 and again lower returns for 1982-1995. Nevertheless, all estimates of the January returns are statistically significant.

The results of Hawawini and Keim (2000, NYSE-AMEX 1962-1994) show that the January effect (relative to the other months) is appr. 11% for small stocks (Decile 10) and 1% for large stocks (Decile 1).

• The usual explanations suggested for the stock market January effect are tax-loss selling (selling loser stocks in December to realize tax-offsetting capital losses in the current year) and institutional window-dressing (selling assets in December not to be included in year-end public disclosures). The empirical evidence is mixed. The January effect also exists in countries without capital gains tax (e.g. Japan before 1989, Canada before 1972). The UK and Australia have January effects, even though their tax years begin on April 1 and July 1. Large (and small) stocks that have risen during the previous year also show January effects. On the other hand, the turn of the year effect in the US seems to have started after introduction of personal taxes in 1917 (Schultz, 1985; Jones, Lee and Apenbrink, 1991). Following the Tax Reform Act 1986 change in fiscal year-end for mutual funds to end-October, empirical evidence suggests a November effect (Bhabra, Dhillon and Ramirez, 1999). Window-dressing would suggest that the January effect is strongest for firms with institutional shareholders, but the evidence suggests larger January effects for firms with individual shareholders (Sias and Starks, 1997).
• Overall problem with these explanations is that there is no substantial negative December effect that reflects downward selling pressure on prices. On the other hand, December trading volumes for loser stocks are high and for winner stocks low (Dyl, 1977), more transactions seem to occur at the (lower) bid price than the (higher) ask price (Keim,1989), more individual investor odd-lot sales than purchases (Dyl and Maberly, 1992).
• The January effect is not an isolated stock market phenomenon. A January effect has also been found in returns and yields of US low rated corporate bonds (see Chang and Pinegar, 1986; Chang and Huang, 1990; Fama and French, 1993; Barnhill, Joutz, and Maxwell, 1997) and municipal bond returns (see Kihn, 1996). Evidence for the government bond and bill market is not so clear. Interest rates on commercial paper that matures across the year-end show a significant higher premium (low price) in the run-up to year end, relative to Treasury bills (Musto, 1997; Griffiths and Winters, 2005).
• At the end of the year, many investors seem to be exiting the financial markets with a preference for liquidity, probably to meet year-end cash flow obligations (Griffiths and Winters, 2005). In fact, money growth is high in December and the money stock falls in January. Moreover, the size of the money changes in December and January appear to be significantly related to the returns on the stock market at that time (Chen and Fishe, 1994 Table 4).

OTHER SEASONAL EFFECTS, CALENDAR ANOMALIES, THE WEATHER, AND OTHER ODDITIES

Empirical studies have turned up a wide range of anomalies relating to seasonality in stock returns. Among these are the Monday effect (Cross, 1973), the Holiday effect (Ariel, 1990), turn-of-the-month effect (Ariel, 1987) and some others. Other anomalies are based on such oddities as the lunar cycle, geomagnetic storms (sunspots), weather (sunshine, rain, cloud cover, temperature), Super Bowl indicators. Many anomalies of the second category are interpreted as supporting a behavioral and psychological approach to stock market returns.

• Fama (1991), Keim (1988) have pointed out that calendar anomalies are anomalies in the sense that asset-pricing models do not predict them, but at the same time most anomalies are irrelevant due to the fact that the abnormal returns observed fall within the limits of relevant bid-ask spreads and therefore do not generate opportunities for arbitrage profits. Why at certain times prices appear to have higher probabilities of being closer to bid or ask prices is subject of further study.
• Jacobsen and Marquering (2008) point out that various weather variables used in stock market anomalies have seasonal patterns closely related to their favorite "Sell in May" or Halloween seasonal dummy (Bouman and Jacobsen, 2002).
• When weather variables are really fundamental driving forces in stock market returns, the effects should be reasonably stable across countries. In fact, when considering countries in the northern hemisphere and southern hemisphere with opposite seasonal patterns in the weather, they find that southern hemisphere countries (of which there are few) exhibit weather effects (temperature and daylight) opposite to what is expected and suggested by the previous literature. Furthermore, countries close to the equator such as Thailand appear to have the largest response coefficients to weather effects, despite the fact that temperature and daylight in these countries have the smallest variability during the year. The conclusion is that weather variables must be proxies for some other global seasonal/calendar pattern in stock returns.
• Goetzmann and Zhu (2002) cannot find evidence of a weather influence on behavior of investors, examined at the same time but located in different regions. They suggest that the weather effect must be limited to traders and market makers working at the location of the exchange. (A cumbersome proposition in today's world of remote access and electronic trading systems.)
• Kelly and Meschke (2007) revisit the widely cited Kamstra et al (2003) study on Seasonal Affective Disorder (SAD) and argue that the observed SAD effect is due to a fault in the original empirical specification (overlapping-dummy Fall and Winter-Fall SAD).
• In another multi-country study, Gregory-Allen, Jacobsen and Marquering (2008) reject the Daylight Saving Time effect reported in previous literature as insignificant, largely due to correction of test-statistics for non-normality in daily returns.
• Ciccone and Etebari (2007) find that in 6 major countries the Sell in May or Halloween effect (May-October) is in fact only a September effect in the large stock indices (However, in this study sample periods are short - post-1985/1991 - and the headline large stock indices do not include dividends). Maberly and Pierce (2004) show that in the U.S. the Sell in May effect disappears from the 1970-1998 data when taking into account the January effect and two large outliers in stock returns, October 1987 and August 1998.
• Psychological studies cited in support of behavioral/psychological influences on stock market investors appear to be mistreated in the anomaly literature (Jacobsen and Marquering, 2008; Kelly and Meschke, 2007). The effect of weather/mood on risk taking is undetermined, the effect of temperature on mood/behavior is examined in studies of extreme temperature differences not relevant for normal investors, weather variables do not necessarily influence behavior when investors work indoor rather than face the weather outdoor. To suggest that psychological studies contribute any evidence for systematic patterns in stock market behavior is a leap of faith.
• (See Keller et al 2005 for a review of the psychological literature relating weather and mood, and the inconsistent relationships.)
• Gerlach (2007) finds that for the U.S. in the period 1980-2003, 5 of 6 statistically significant anomalies examined (turn-of-the-month effect, Fall effect or Halloween effect, lunar cycle effect, rain effect, temperature effect) disappear when taking into account the days with major (macro) economic news announcements. The Holiday effect suffers the same fate, but is referred to only in a footnote due to the fact that the anomaly is not statistically significant from the start. The January effect is weakened by eliminating news days, but is the only anomaly to survive.
• Thus, statistically significant calendar and weather anomalies are not caused by market psychology or institutions but reflect a process of data mining where certain variables happen to coincide with days of news that is important for stock markets. Using days without announcements, there is no evidence for calendar anomalies.
• Correlation does not imply causality - examples: http://pbil.univ-lyon1.fr/members/lobry/corcau
• Datasnooping: http://data-snooping.martinsewell.com/
• "You need a plausible explanation for why the indicator should work. Without such an explanation, then you run a big risk that the indicator is based on nothing more than a fluke of the data. My favorite example of why this is so comes from David Leinweber, a visiting faculty member in CalTech's economics department. Several years ago, wanting to illustrate the perils of mining the data for spurious correlations, he searched through all the data on a United Nations CD-ROM to find the indicator with the most statistically significant correlation with the S&P 500. His discovery: butter production in Bangladesh." [Hulbert, 2005; Is January special after all? MarketWatch.com]

METHODOLOGICAL PROBLEMS AND RETURN BIASES

See the other page on data problems and return measurement problems.

Arguably, the main issue in (stock) return anomaly studies is the assumption made on the 'normal' return. Historically, the constant mean, market adjusted or CAPM-beta model have been used to define the 'normal' stock return that is assumed to capture the effect of different risk on return. Besides the fact that the CAPM-beta model is usually applied in a casual and incorrect way (incorrect beta estimates due to high-frequency data or incorrect 'market' measurement), we know that the simple textbook risk model is inadequate. Various other elements of real world risk need to be incorporated also. The, arguably, modern model of risk adjustment includes the Fama-French risk factors (beta, SMB, HML) and liquidity risk, return skewness, and macro economic risk (possibly the fundamentals behind the SMB and HML effect).